Bayes Club Bayes
club is an informal meeting of people interested in Bayesian
statistics. Our aim is to exchange ideas and recent developments in our
research.
Meetings are held on Fridays, from 16:00 (new starting time), at VU University Amsterdam (Science Building) in room P6.56.
Meetings are held on Fridays, from 16:00 (new starting time), at VU University Amsterdam (Science Building) in room P6.56.
Upcoming
talks
- May 25, 2012 - Botond Szabó - Adaptive Bayesian techniques for inverse problems (abstract)
- June 15, 2012 - Catia Scricciolo - tba
Recent
talks
- May 11, 2012 - Suzanne Sniekers - Credible sets in the fixed design model with Brownian motion prior (abstract)
- April 27, 2012 - Eduard Belitser - On Bayesian construction of exact confidence sets (abstract)
- April 20, 2012 - Johannes Schmidt-Hieber - Posterior concentration in high-dimensional regression under sparsity (abstract)
- December 16, 2011 - Eduard Belitser - On estimation of high dimensional vector of binomial proportions (abstract)
- November 25, 2011 - Harry van Zanten - A differential equations approach to nonparametric Bayesian drift estimation for diffusions on the circle (abstract)
- September 16, 2011 - Tim van Erven - Bayesian Methods in Online Learning (abstract)
- June 17, 2011 - Frank van der Meulen (abstract)
- June 10, 2011 - Subhashis Ghoshal - Predicting Proportion of False Discovery Proportions in Dependent Multiple Tests
- May 13, 2011 - Haralambie Leahu - On the BvM phenomenon in the Gaussian white noise model (abstract)
- April 8, 2011 - Bartek Knapik - Semiparametric posterior limits under local asymptotic exponentiality (abstract)
- April 1, 2011 - Bas Kleijn - Bayesian efficiency in mixture models
- March 18, 2011 - Aad van der Vaart (abstract)
Abstracts
In my presentation I will talk about adaptive Bayesian techniques, such as empirical and hierarchical Bayes method, and apply them to solve mildly ill-posed inverse problems. I will investigate the behaviour of the adaptive posterior distribution from a frequentist point of view and show that the posterior distribution achieves the minimax rate of contraction up to a slowly varying term. Furthermore, if time allows, I will examine how much confidence can we put in the adaptive credible sets and try to construct the largest set on which we can trust adaptive credible sets as a measure of uncertainty. This is ongoing joint work with Aad, Harry, and Bartek.
We will consider the problem of estimating an unknown function f at a given point in a fixed design setting. A Bayesian approach with Brownian motion prior will be used to derive an estimator, which we will study using frequentist methods. We will investigate how the bias depends on the Hölder smoothness of the function f. This result will be used to study the coverage of Bayesian credible sets for this model.
We consider one general, simple theorem which gives a recipe how to construct exact (i.e., non-asymptotic) confidence sets under certain conditions. Next we discuss some applications.
Recently, highdimensional problems and sparsity constraints have gained a lot of attention due to their applicability in biology. In this talk, we consider a fully Bayesian approach to high-dimensional regression. For a class of priors, which accounts for sparsity, we provide results for the contraction rate of the posterior. We discuss the assumptions on the design matrix and relate them to existing work. For specific situations, we are able to determine the behavior of credible intervals. This is ongoing joint work with Aad and Ismael.
We consider the problem of (minimax and oracle) estimation of high (infinite) dimensional vector of binomial proportions. Under some conditions we derive the asymptotic behavior of the minimax risk over some nonparametric classes, in particular, a "binomial version" of Pinsker's result. Further, we might touch upon the issue of (empirical) Bayesian adaptation and the problem of optimal allocation of observations.
I will report on a joint project with Andrew Stuart and Yvo Pokern in which we study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. We specify a centered Gaussian prior on the drift with a precision operator that is of differential form. It is proved that the posterior is Gaussian as well and we give an explicit expression for the posterior precision operator and show that the posterior mean is the solution of a certain differential equation. Moreover, we bound the rate at which the posterior contracts around the true drift function. The results rely on tools from the analysis of differential equations and new functional limit theorems for the local time of diffusions on the circle.
I will give an introduction to online learning, which deals with
decision problems that can be formulated as a repeated game between the
statistician and an adversary. This is a natural way to model problems
like spam detection and optimization of financial portfolio's, which
have an adversarial component. But there are also applications to data
compression, for which particularly strong performance guarantees are
possible.
Many standard algorithms in online learning can be interpreted as Bayesian methods, or approximations thereof. I will work out several examples. Time permitting, I will also present recent work with Peter Grünwald, Wouter Koolen and Steven de Rooij, in which we analyse a new algorithm by viewing it as an approximation to Bayes, and show that fast convergence of the Bayesian posterior implies better decisions.
Many standard algorithms in online learning can be interpreted as Bayesian methods, or approximations thereof. I will work out several examples. Time permitting, I will also present recent work with Peter Grünwald, Wouter Koolen and Steven de Rooij, in which we analyse a new algorithm by viewing it as an approximation to Bayes, and show that fast convergence of the Bayesian posterior implies better decisions.
I will review some methods to deal with MCMC on spaces of varying dimension, in particular the reversible jump algorithm by Green.
As an application, I will show how this method can be used to estimate the drift of a discretely observed diffusion. The particular hierarchical prior that we propose requires a slight adaptation of the basic algorithm. This concerns joint work with Moritz and Harry.
As an application, I will show how this method can be used to estimate the drift of a discretely observed diffusion. The particular hierarchical prior that we propose requires a slight adaptation of the basic algorithm. This concerns joint work with Moritz and Harry.
One of the most interesting consequences of the (parametric) BvM Theorem is that the Bayesian and frequentist distributions of the estimation error are asymptotically the same, having asymptotic Gaussian shape; in particular, Bayesian credible sets and frequentist confidence regions must coincide asymptotically. This talk investigates the occurrence of this "phenomenon" in the Gaussian white noise model.
I will first give a 15 minutes presentation on Bayesian inverse problems, as preparation for the Philips Award session during NMC 2011 in Enschede. Then I will talk about semiparametric posterior limits under the condition of local asymptotic exponentiality. Consider the model indexed by a real-valued parameter \theta and a nuisance parameter \eta. Every element of the model has a density p_{\theta,\eta}(x) given by \eta(x-\theta), where \eta is a density function, supported on positive reals, and \eta(0) is positive and finite. I will show some results on the asymptotic behavior of the marginal posterior for the parameter of interest \theta in this semiparametric model.
My
intention is to review some computational methods for posterior based
on Gaussian priors: expectation propagation and Laplace approximation.
The first I learned (mostly) from a thesis in Nijmegen, regarding
the
second I review the paper on INLA in JRSSb2009(?) by Chopin, Rue et al.
They are claimed to be much faster than MCMC and just as good/bad.
No new stuff, and no theory, but if I find the time I'll prepare some pictures, and there may be possibilities of implementing similar algorithms on other models of interest.
No new stuff, and no theory, but if I find the time I'll prepare some pictures, and there may be possibilities of implementing similar algorithms on other models of interest.
If you have any comments, questions, requests, suggestions etc. please contact me - see my website for contact information.
Bartek Knapik - 20/05/2012